Admixture?gene flow between previously isolated populations?has shaped the genomes of almost all human populations. The mosaic ancestry of admixed populations varies along their genomes and between individuals, leading to differences in phenotypic variation and disease risk. Indeed, admixture is one of the fastest evolutionary processes to dramatically change the composition of a population. Yet, methods to study the evolutionary history of admixed populations are lacking. Admixed populations are often considered as simple linear combinations of their sources, missing variation in mating patterns and migration rates over time. Over the next five years, the Goldberg lab will combine quantitative methods development with empirical data analysis to addresses two fundamental components of the evolutionary history of admixed populations: 1) inference of demographic history and its consequences for neutral and deleterious patterns of variation, and 2) characterization of loci under positive selection before and after admixture. Specially, we will extend our mechanistic model of sex-specific admixed to allow for non-random mating patterns empirically observed in admixed populations. We will apply this model to infer the demographic history of Cape Verdeans, with Portuguese and West African ancestry, testing for differences in mating patterns between islands that differ in their size and colonization history. Under this model, we will also use Cape Verde as a case study of the relationship between demography and deleterious variation in admixed populations. Next, we will develop new statistics to infer positive selection using admixed genomes both before and after admixture. Admixture can both mask classic signatures of positive selection, and provide new genetic material upon which selection can act. Therefore, methods that account for the specific demographic and selective signatures of admixed populations are necessary. We will incorporate these statistics into a simulation-based learning method to make it easily accessible. Finally, we will apply our new statistics to move beyond identification of putatively selected loci to characterize the strength and timing of selection in multiple admixed African diaspora populations in different environments. This proposal enhances our understanding of evolutionary processes in admixed populations, produces usable statistical methods, and elucidates the population and selective history of diverse global populations. The Goldberg lab is uniquely positioned to accomplish these goals because of our experience combining exact mathematical models with realistic inference frameworks to study the processes generating genetic variation in admixed groups.